Grover's algorithm

Grover's algorithm isquantum algorithmsearching an unsorted databaseN entriesO(N^1/2) timeO(log₂N) space. It was invented by Lov Grover1996.

Classically, searching an unsorted database requireslinear search, whichO(N). Grover's algorithm provides "only" quadratic speedup, unlike other quantum algorithms whichthoughtprovide exponential speedup over their classical counterparts. However,isfastest possible quantum algorithmsearching an unsorted database,even quadratic speedupconsiderable when Nlarge.

Like all quantum computer algorithms, Grover's algorithmprobabilistic insense thatgivescorrect answerhigh probability. The probabilityfailure can be decreased by repeatingalgorithm.

Although Grover's algorithmoften described as searchingdatabase,would be more accuratedescribeas invertingfunction. Iffunction y=f(x) can be computed by an algorithm forquantum computer, then Grover's algorithm can calculate x, given y. Grover's algorithm wouldn't typically be usedsearchnames inphone book, butcould be usedsearch forkey that decrypts an encrypted message. Iffunctioncomputing f(x)given asblack box, then Grover's algorithm isfastest possible algorithminverting it.

Grover's algorithm can be usedmeanmedian estimation,solvingcollision problem. It can be usedsolve NP-complete problems by performing exhaustive searches over all possible solutions. This will resultconsiderable speedups over classical methods, but won't achieve polynomial runtimeNP-complete problems.

Below, we presentbasic formGrover's algorithm, which searches forsingle matching entry. The algorithm can be further optimized if theremore than one matching entry andnumbermatchesknown beforehand.

Procedure

Consider an unsorted databaseN entries. The algorithm requires an N-dimensional state space H, which can be supplied by log₂N qubits.

Numberdatabase entries 0, 1, ... N-1. Choose an observable Ω acting on HN distinct eigenvalues. Byspectral theorem, we can construct an orthonormal basiseigenkets {|0⟩,|1⟩,...|N-1⟩} (see bra-ket notation). The eigenvalues {λ₀,λ₁,...,λ_N-1} label distinct database entries. We wishfindlabel |ω⟩ fordatabase entry matching our search criterion.

Weprovided withunitary operator U_ω which compares database entriesour search criterion. The algorithm does not specify how this subroutine works, butmust bequantum subroutine that workssuperpositionsstates,it must havefollowing effects onlabel kets:

U_ω |ω⟩ = - |ω⟩

U_ω |x⟩ = |x⟩     x ≠ ω

The steps ofGrover algorithm are:

1. Initializesystem tostate

   |s⟩ = N^-1/2 Σ_x |x⟩

2. Performfollowing "Grover iteration" r(N) times.
   r(N)described below;N>>1, r ≈ πN^1/2/4.
   1. 1. Applyoperator U_ω
      2. Applyoperator U_s = 2 |s⟩ ⟨s| - I
3. Performmeasurement Ω. The measurement result will be λ_ωprobability approaching 1N>>1. From λ_ω, ω may be obtained.

Explanation ofAlgorithm

Our initial state is

|s⟩ = N^-1/2 Σ_x |x⟩

Considerplane spanned by |s⟩|ω⟩. Let |ω^×⟩ beketthis plane perpendicular|ω⟩. Since |ω⟩one ofbasis vectors,overlap is

⟨ω|s⟩ = N^-1/2

In geometric terms, therean angle (π/2 - θ) between |ω⟩|s⟩, where θgiven by:

cos (π/2 - θ) = N^-1/2

sin θ = N^-1/2

The operator U_ω isreflection athyperplane orthogonal|ω⟩;vectors inplane spanned by |s⟩|ω⟩,acts asreflection atline through |ω^×⟩. The operator U_s isreflection atline through |s>. Therefore,state vector remains inplane spanned by |s⟩|ω⟩ after each applicationU_safter each applicationU_ω,itstraighforwardcheck thatoperator U_sU_ωeach Grover iteration step rotatesstate vector by an angle2θ toward |ω⟩.

We needstop whenstate vector passes close|ω⟩; after this, subsequent iterations rotatestate vector away from |ω⟩, reducingprobabilityobtainingcorrect answer. The numbertimesiterategiven by r. In orderalignstate vector exactly|ω⟩, we need:

π/2 - θ = 2θr

r = (π/θ - 2)/4

However, r must be an integer, so generally we can only set rbeinteger closest(π/θ - 2)/4. The angle between |ω⟩ andfinal state vectorO(θ), soprobabilityobtainingwrong answerO(1 - cos²θ) = O(sin²θ).

For N>>1, θ ≈ N^-1/2, so

r ≈ π N^1/2 / 4

Furthermore,probabilityobtainingwrong answer becomes O(1/N), which goeszerolarge N.

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References:

• Grover L.K.: A fast quantum mechanical algorithmdatabase search, Proceedings, 28th Annual ACM Symposium onTheoryComputing, (May 1996) p. 212
• http://www.bell-labs.com/user/feature/archives/lkgrover/